nLab
subcanonical coverage

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Topos Theory

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Definition

A coverage (resp. Grothendieck topology, resp. Grothendieck pretopology) defining a site is called subcanonical if all representable presheaves on this site are sheaves. Of course, a subcanonical site is one whose coverage is subcanonical.

The term “subcanonical” comes about because the largest coverage for which the representables are sheaves is called the canonical coverage, and the subcanonical coverages are precisely the “sub-coverages” of the canonical one.

Effective-epimorphic sieves

An alternate definition is that a Grothendieck coverage is subcanonical if and only if all of its covering sieves RC(,U)R\hookrightarrow C(-,U) are effective-epimorphic, meaning that the morphisms f:VUf:V\to U in RR form a colimit cone under the diagram consisting of all morphisms between them over UU. To see this, first recall that if RC(,U)R\hookrightarrow C(-,U) is a sieve, then a functor X:C opSetX:C^{op}\to Set satisfies the sheaf axiom for RR if and only if

  • for every family (x f) fR(x_f)_{f\in R} which is compatible, in the sense that X(g)(x f)=x fgX(g)(x_f) = x_{f g} whenever this makes sense, there exists a unique xX(U)x\in X(U) such that x f=X(f)(x)x_f = X(f)(x).

Interpreting this when XX is a representable functor C(,Z)C(-,Z), we obtain

  • for every family of maps (h f:VZ)(h_f:V\to Z), where f:VUf:V\to U is in RR, such that h fg=h fgh_f g = h_{f g} for any g:VVg:V'\to V, there exists a unique k:UZk:U\to Z such that h f=kfh_f = k f.

But this says precisely that RR is effective-epimorphic, as defined above.

In fact, since the covering sieves in a subcanonical coverage must also satisfy pullback-stability, they must be not only effective-epimorphic but universally effective-epimorphic (meaning that any pullback of them is effective-epimorphic). It is then easy to see that the canonical coverage consists precisely of all the universally effective-epimorphic sieves.

Note also that if f:VUf:V\to U is a single morphism having a kernel pair p,q:V× UVVp,q:V\times_U V \;\rightrightarrows\; V, then the sieve generated by ff is effective-epimorphic iff ff is the coequalizer of its kernel pair, and thus iff ff is a effective epimorphism.

Revised on November 26, 2013 23:46:15 by Urs Schreiber (77.251.114.72)